Contributions to Algebra and Geometry Volume 45 (2004), No. 1, 117-131.
Normalizing Extensions of Semiprime Rings
Miguel Ferrero1 Rog´erio Ricardo Steffenon2 Instituto de Matem´atica, Universidade Federal do Rio Grande do Sul
91509-900 Porto Alegre - RS, Brazil e-mail: [email protected]
Centro de Ciˆencias Exatas e Tecnol´ogicas, Universidade do Vale do Rio dos Sinos 93022-000 S˜ao Leopoldo - RS, Brazil
e-mail: [email protected].
Abstract. In this paper we study normalizing extensions of semiprime rings.
For an extension S of R we construct the canonical torsion-free S∗, which is a normalizing extension of the symmetric ring of quotientsQofR. We extend results which are known for centralizing extensions and for normalizing bimodules to one- to-one correspondence between closed ideals. Finally we study prime ideals, non- singular prime ideals and (right) strongly prime ideals of intermediate extensions.
MSC 2000: 16D20, 16D30, 16S20, 16S90
Introduction
Prime ideals in ring extensions R ⊆ S have extensively been studied in recent years. In particular, when the extension is generated by a finite set of R-centralizing elements, S is called a liberal extension ([12], [13]). A normalizing extension is again a finite extension which is generated by a set of R-normalizing generators ([7], [8], [9], [11]). Also strongly normalizing extensions have been considered in [10].
1Partially supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq, Brazil).
2Supported by a fellowship granted by Universidade do Vale do Rio dos Sinos (UNISINOS, Brazil). Par- tially supported by Funda¸c˜ao de Amparo `a Pesquisa do Estado do Rio Grande do Sul (FAPERGS, Brazil).
Some results of this paper are contained in the Ph.D. thesis written by the second named author and presented to Universidade Federal do Rio Grande do Sul (Brazil) [14].
0138-4821/93 $ 2.50 c 2004 Heldermann Verlag
Prime ideals in (not necessarily finite) centralizing extensions were studied in [1], [2] and [3]. In those papers the results on prime ideals were obtained as applications of the results on closed submodules of centralizing bimodules over prime and semiprime rings.
Recently the results on closed submodules were extended to normalizing bimodules over semiprime rings in [5]. The main result in that paper gives a one-to-one correspondence between closed submodules of a normalizing bimodule M over a semiprime ring R, closed submodules of its extension to a bimodule M∗ over the symmetric ring of quotients Q of R and closed submodules ofM0, the set of all theR-normalizing elements of M∗ (actually, this last set is not a module, but it can be treated in a very similar way over the set of all the R-normalizing elements of Q).
The purpose of this paper is to extend the results on centred extensions to normalizing extensions, applying the results of [5]. Throughout the paper R is a semiprime ring and S is a normalizing extension of R. Recall that if M is an R-bimodule, then M is said to be an R-normalizing bimodule if there exists X = (xi)i∈Ω ⊆ M such that M is generated over R by the setX and Rxi =xiR, for everyi∈Ω. A ring S is said to be a normalizing extension of R if R ⊆S and S is a normalizing bimodule over R.
In Section 1, we consider some types of normalizing extensions and give examples showing that they are, in general, all different. We show that the torsion submodule is not in general an ideal, but it is an ideal if the extension is of some special type, called essentially normalizing extension. The canonical torsion-free extension S∗ of S is constructed in Section 2.
In the next Section 3 we extend the results on closed submodules of [5] to closed ideals. In Section 4 we study intermediate extensions and we show that the one-to-one correspondence can be extended to this context. We also prove that the correspondence preserves prime and semiprime ideals of intermediate extensions.
Finally, in Section 5 we study strongly prime and non-singular prime ideals of interme- diate extensions of prime rings. We extend here several results of ([2], Section 6).
In the paper we use freely the terminology and results of [5]. In particular, unless oth- erwise stated, submodule means sub-bimodule. An ideal H of R is always a two-sided ideal and this will be denoted by HR. The set of all essential ideals of R is denoted by E(R) and we will write simply E if there is no possibility of misunderstanding. The symmetric Martindale ring of quotients of R will be denoted byQ. The right annihilator of a subset F inR will be denoted by AnnR,r(F). The notations ⊂ and ⊃ mean strict inclusions.
1. Normalizing extensions of rings
LetR be a semiprime ring and S a ring extension of R. Recall that an elementx∈S is said to be centralizing (resp. normalizing) over R if rx =xr, for every r ∈ R (resp. Rx = xR).
Also,x is said to be strongly normalizing over R if Ix=xI, for any ideal I of R [10].
Another notion we will use in the paper is the following
Definition 1.1. An element x ∈ S is said to be essentially normalizing if x is normalizing over R and satisfies the following condition: for any I ∈ E there exists J = J(I) ∈ E such that J x⊆xI and xJ ⊆Ix.
The ring extension S ⊇ R is said to be a centralizing (resp. normalizing, strongly normal- izing, essentially normalizing) extension if there exists a subset X = (xi)i∈Ω of S which is centralizing (resp. normalizing, strongly normalizing, essentially normalizing) over R and S =Pi∈ΩRxi.
Consider the following conditions:
(i) S is a centralizing extension of R.
(ii) S is a strongly normalizing extension ofR.
(iii) S is an essentially normalizing extension ofR.
(iv) S is a normalizing extension ofR.
We immediately have
Lemma 1.2. The following implications hold: (i)⇒(ii)⇒(iii)⇒(iv).
Now we give examples to show that the converse of the above implications do not hold.
Example 1.3. Assume that φ is an involution of a field K and put S = K[x, φ], the skew polynomial ring over K (we write the coefficients in the right). Then S is a strongly normalizing extension of K which is not centralizing over K. This is clear since s ∈ S is
centralizing if and only if s∈Pix2iK. 2
Example 1.4. Let Z be the ring of integers, R = Z×Z, φ : R → R the automorphism defined by φ(n, m) = (m, n), for n, m ∈ Z. Put S = R[x, φ] the skew polynomial ring over R. An element s ∈ S is R-normalizing if and only if either s ∈ Pni=0x2iR (in this case s is centralizing overR) or s∈Pni=0x2i+1R. It is easy to see that in the last case s is essentially normalizing, but not strongly normalizing. It follows that S is an essentially normalizing
extension of R and it is not strongly normalizing. 2
Example 1.5. Let Z2 be the prime field of characteristic 2 and R = Z2 ×Q∞i=2Zi, where Zi = Z for all i ≥ 2, the product of rings. Consider the epimorphism of R given by φ( ¯a1, a2, a3, . . .) = ( ¯a2, a3, . . .), where ¯a = a+ 2Z ∈ Z2, and put S = R[x, φ]. Then S is a normalizing extension ofRwith (xi)i≥0 as a set of normalizing generators, but is not essen- tially normalizing. In fact, ifS is essentially normalizing over R, then at least one generator of S has to be of the form s = r0 +r1x+· · ·+rnxn, where r1 = (¯1, r12, r13, . . .). We have that I =Z2×Q∞i=22Z ∈ E(R) and do not exist J ∈ E(R) with J s⊆sI. 2 The torsion submodule T(M) of a normalizing bimodule M over R has been defined and studied in ([5], Sections 2 and 5). The normalizing bimodule M is said to be torsion-free if T(M) = 0. When we consider a normalizing extension S of R it would be convenient that torsion submodule T(S) will be an ideal. However, this is not the case, as we will see in the following example.
LetR=K[x1, x2, . . .] be the ring of polynomials in an infinite set of indeterminates{x1, x2,. . .}
over a fieldK. ThenR is a commutative prime ring. Consider theK-endomorphismsσ and φ of R defined byσ(xi) =xi+1 for any i,φ(xi) =xi−1 if i≥2 and φ(x1) = 0.
LetW be the free monoid generated by X, Y and consider the ring defined as follows:
T =R < X, Y, σ, φ >={X
w
waw :aw ∈R, w ∈W},
where Pwwaw denotes a finite sum. The addition in T is defined as usual and the multipli- cation by xiX =Xσ(xi) and xiY =Y φ(xi).
A monomial is an element of the typewa ∈T, where a is a monomial in R and w∈W. ThusT can be regarded as a graded ring in two different ways:
(Gr1) The degree of a monomial is the sum of the degrees with respect to X and Y.
(Gr2) The degree of the monomial wa is the usual total degree of the coefficient a inR with respect to{x1, x2, . . .}.
LetI = (Xx1) be the ideal of T generated by Xx1. ThenI is homogeneous in both gradua- tions because the generator is Gri-homogeneous, i= 1,2.
Lemma 1.6. Under the above notation we have:
(i) X, Y /∈I.
(ii) If 06=f ∈R, then f /∈I and XY f /∈I.
Proof. (i) Assume that X ∈ I. Then X = PjfjXx1gj, where fj, gj ∈ T. Computing the Gr2-degree we obtain a contradiction. Similarly Y /∈I.
(ii) It is enough to show that XY f /∈ I. Suppose that 0 6= f ∈ R and XY f ∈ I. Then XY f = PjfjXx1gj, where fj, gj ∈ T. Since W is a free monoid there exists at least one j for whichfj contains a summand beginning with XY. UsingGr1 we obtain a contradiction.
2
Put S = T /I and denote by x =X+I, y = Y +I. It is clear that S is a ring extension of R. We have
Example 1.7. S is a normalizing extension of the prime ring R with normalizing generator set (w+I)w∈W. Moreover, x, y ∈ T(S) and xy /∈ T(S). In particular, T(S) is not an ideal of S.
Proof. It is clear that Rx = xR and Ry = yR. The first part follows. Since R is a commutative prime ring, xx1 = 0 andx1y= 0, we obtainx, y ∈T(S). Alsoxixy=xxi+1y= xyxi, for all i, so xy is R-centralizing. If xy ∈ T(S), then for some 0 6= r ∈ R we have
xyr = 0, i.e., XY r∈I, a contradiction. 2
Note that a torsion-free normalizing extensionS of R is an essentially normalizing extension ([5] Lemma 2.4). In this caseT(S) = 0 is, of course, an ideal. We show that the same is true for an essentially normalizing extension.
Assume thatS is a normalizing extension ofR andX = (xi)i∈Ω is a set ofR-normalizing generators. We may suppose that xi0 = 1, for some i0 ∈Ω.
IfI ⊆P are ideals of S, as in [5] we define the closure of I inP by [I]P ={x∈P : there exist F, H ∈ E such thatF xH ⊆I}.
The ideal I is said to be closed in P if [I]P =I.
Lemma 1.8. If S is an essentially normalizing extension of R and I ⊆ P are ideals of S, then [I]P is also an ideal of S.
Proof. We already know that [I]P is a submodule of RSR, by the results in [5]. Let y∈[I]P and take any generator x ∈ X. Then there exist F, H ∈ E such that F yH ⊆ I. Also, by Definition 1.1 there exist F0, H0 ∈ E such that F0x ⊆ xF and xH0 ⊆ Hx. So we have that F0xyH ⊆ xF yH ⊆xI ⊆I and F yxH0 ⊆ F yHx⊆Ix ⊆I. Therefore xy, yx∈[I]P and the
result follows. 2
As a particular case of the above we have the following
Corollary 1.9. Assume that S is an essentially normalizing extension of R. Then the torsion submodule T(S) is an ideal ofS.
Proof. By Lemma 1.8, T1(S) = [0]S is an ideal. The result follows since T(S) = T2(S) =
[T1(S)]S ([5], Section 5). 2
In general, to define the torsion ideal of S we extend the definition of T(S). In this paper we want to study closed ideals. So we may assume that the set of closed ideals of S is not empty. Thus we define the torsion ideal t(S) of S as the intersection of all the closed ideals of S, i.e., the smallest closed ideal. Thus any closed ideal of S contains t(S). It is easy to see that there is a one-to-one correspondence between the closed ideals of S and the closed ideals of S/t(S), via the canonical projection (see [5], Lemma 2.2).
There is another way to define t(S). We put t1(S) = (T(S)), the ideal generated by T(S). Let t2(S) be the ideal generated by t02(S), where t02(S) is the submodule of S such that t02(S)/t1(S) = T(S/t1(S)). We proceed similarly to define tγ(S) for any non-limit ordinal number γ. If α is a limit ordinal we put tα(S) =Pβ<αtβ(S). We have a transfinite sequence of ideals. Therefore there exists an ordinalρwithtρ(S) =tρ+1(S). This means that T(S/tρ(S)) = 0 and hence tρ(S) is a closed ideal ofS. It is easy to see that t(S) =tρ(S).
Remark 1.10. If S is an essentially normalizing extension of R, then t(S) = T(S), by Corollary 1.9.
Example 1.11. In Example 1.7 the torsion ideal of S is the ideal generated by xand y and the factor ring S/t(S) = R.
2. The canonical torsion-free extension
IfM is a normalizing bimodule overR, then there exists a canonical torsion-free normalizing Q-bimoduleM∗ ofM, where Q is the symmetric ring of quotients of R ([5], Section 8). The purpose of this section is to show that the same is true for a normalizing extension.
Let S =Pi∈ΩRxi be a normalizing extension of R with normalizing generators (xi)i∈Ω. First we assume thatS is torsion-free.
Let W denote the free monoid generated by Ω (with the empty word as identity). Take w =i1i2. . . in ∈W and consider xw =xi1 ·. . .·xin ∈ S, a normalizing element of S. Using similar notation as in ([5], Section 3), we put Aw =AnnR,r(xw) , Bw = AnnR,l(xw) and let φw be the isomorphism of ringsφw :R/Bw →R/Aw defined by φw(r+Bw) =r0+Aw, where rxw =xwr0.
For any w ∈ W we consider a free R/Aw-module Tw = wR/Aw, with the unitary basis w, and define a structure of left R/Bw-module on Tw by (r+Bw)w =wφw(r+Aw), r∈R.
Then Tw is an R-bimodule and T = Lw∈WwR/Aw is a normalizing bimodule over R with (w)w∈W as a set of normalizing generators. Moreover, we see thatT has a ring structure and so is a normalizing extension of R.
In fact, if wrw and vrv are monomials in T we put wrw·vrv = wvrwvrv, where rwv is an element ofRsuch thatrwxv =xvrwv. It is easy to see that this is a well-defined multiplication and defines a ring structure on T. Also we may considerR ⊆T via the application sending r to r·1T and so T is a normalizing extension of R. Moreover, the application Φ : T → S defined by Φ(w) = xw is an epimorphism of normalizing extensions such that Φ|R=idR.
Note thatT is torsion-free overR. Thus by Lemma 2.2 of [5] there is a canonical one-to- one correspondence between the set of all the closed ideals of S and the set of all the closed ideals of T containing KerΦ.
Now we define an extension of T to a normalizing Q-bimodule T∗, as in ([5], Section 3). Aw and Bw are closed ideals of R and so there exist closed ideals A∗w and Bw∗ of Q with A∗w∩R=AwandBw∗∩R=Bw. Thus the isomorphismφw can be extended to an isomorphism from Q/Bw∗ to Q/A∗w, denoted by φw again ([5], Corollary 1.2). Put T∗ = Lw∈WwQ/A∗w, the canonical extension of T to a bimodule T∗ over Q. Note that AnnQ,r(w) = A∗w and AnnQ,l(w) = Bw∗. It is easy to see that T∗ is a ring extension of T and a torsion-free normalizing extension of Q, with (w)w∈W as a set of normalizing generators. Also, for any x∈T∗ there exists H ∈ E such thatxH ⊆T and Hx⊆T.
Note that the construction ofT∗ here is similar to the construction ofM∗ in ([5], Section 3). In fact, to see this it is enough to consider the extension S as generated over R by (xw)w∈W instead of (xi)i∈Ω. Thus we may apply the results of that paper. In particular, by ([5], Theorem 4.9) there is a one-to-one correspondence via contraction between the set of all the R-closed submodules of T and the set of all the Q-closed submodules of T∗. We have Lemma 2.1. The one-to-one correspondence above is a one-to-one correspondence between closed ideals.
Proof. LetIbe anR-closed submodule ofT andI∗the extension ofIto aQ-closed submodule of T∗. IfI∗ is an ideal so isI =I∗ ∩T. Conversely, if I is an ideal of T, s∈I∗ and y∈T∗, then there existH, F ∈ E such that sH ⊆I,Hs⊆I,yF ⊆T and F y ⊆T, by Corollary 4.8 of [5]. Hence F ysH ⊆ I and HsyF ⊆ I, and so sy ∈ I∗ and ys ∈ I∗, by the above quoted
result. 2
We consider again the epimorphism Φ :T → S. Since S is torsion-free over R, 0 is a closed ideal of S. Then KerΦ is a closed ideal of T and so there exists a Q-closed ideal K∗ of T∗ such that K∗∩T = KerΦ. We put S∗ = T∗/K∗ and denote by j :S →S∗ the application defined as follows. Let x ∈ S and take y ∈ T such that Φ(y) = x. Thus y ∈ T∗ and
we put j(x) = π(y), where π : T∗ → S∗ is the canonical projection. Since K∗ is a closed ideal of T∗ the ring S∗ is a torsion-free normalizing extension of Q and j is an injective ring homomorphism, called the canonical injection (cf. [5], Section 4).
Now let S be any normalizing extension of R. We consider the torsion-free normalizing extension of S/t(S), where t(S) is the torsion ideal of S. Then there exists the torsion-free extension of S/t(S), denoted again by S∗. We define j : S → S∗ as the composition of the canonical mappings S →S/t(S) and S/t(S) → S∗, where the second application is the canonical injection.
Definition 2.2. The pair (S∗, j) is called the canonical torsion-free extension of S to a normalizing extension of Q.
It is clear that Ker(j) = t(S) and we may consider S ⊆ S∗ if and only if S is torsion-free.
If (xi)i∈Ω is a set of R-normalizing generators of S, then (j(xi))i∈Ω is a set ofQ-normalizing generators ofS∗. Finally, the pair (S∗, j) satisfies a universal property, as in ([5], Section 8), so is unique up to isomorphisms.
Note that since S∗ is torsion-free, thenS∗ is always an essentially normalizing extension of Q.
Example 2.3. Let S be the normalizing extension given in Example 1.7. Then S∗ = Q, since S/t(S) = R. We can modify the example in such a way that the canonical extension is not trivial. For example, if S1 =S[Y], where Y is a set of indeterminates, then S1∗ =Q[Y] is a polynomial ring over Q.
3. The one-to-one correspondence
A one-to-one correspondence between closed submodules is obtained in ([5], Theorem 8.3).
The purpose of this section is to show that the same is true for ideals of normalizing exten- sions.
In [5], the set Z of all the R-normalizing elements of Qwas considered (Section 1). This is a multiplicative semigroup with an identity and has an addition partially defined, but is not in general a ring. Also in Sections 6–8 of that paper, the subset M0 of all the elements of M∗ which are R-normalizing plays an important role. This subset has also partially defined addition and a multiplication by the elements ofZ, and the operations have natural properties. We extended the terminology by saying that M0 is a Z-module.
We consider here the corresponding sets. Let S0 ={x ∈ S∗ : Rx = xR}. Hence S0 is a semigroup with identity element and is a Z-module in the above sense. Also the operations have natural properties such as associativity and distributivity when addition is defined. We say here thatS0 is a Z-semigroup. Note that by definitionZ ⊆S0.
We can consider semigroup ideals ofS0. A semigroup idealIis a subset with the property:
for anyx∈I ands∈S0 we havesx, xs ∈I. Actually a more restrictive concept is of interest here:
Definition 3.1. A semigroup (submodule, semigroup ideal) I of S0 is said to be saturated if the following holds: if a1, a2, . . . , an ∈I and the addition a1 +a2+. . .+an is defined in S0, then a1+a2+. . .+an ∈I.
If A and B are saturated ideals of S0, we define the product ofA and B by AB={
n
X
i=1
aibi :ai ∈A, bi ∈B and
n
X
i=1
aibi ∈S0}.
It is easy to see that AB is also a saturated ideal of S0.
Recall that an ideal H of Z is said to be essential if AnnZ(H) = 0. The set of all the essential ideals of Z is denoted by E(Z). An ideal I of S0 is said to be closed if s ∈S0 and sH ⊆I, for some H ∈ E(Z), implies that s∈I.
Note that S, S∗ and S0 are defined as in ([5], Theorem 8.3). The one-to-one correspon- dence in that theorem also shows that any closed submodule of S0 is saturated. Hence any closed ideal of S0 is a saturated ideal.
As an easy consequence we have
Theorem 3.2. LetS be a normalizing extension of a semiprime ringR, (S∗, j)the canonical torsion-free extension of S and S0 the normalizer of R in S∗. Then there is a one-to-one correspondence between the set of all theR-closed ideals ofS, the set of all theQ-closed ideals of S∗ and the set of all the Z-closed ideals of S0. Moreover, the correspondence associates the closed ideal I of S with the closed ideal I∗ of S∗ and the closed ideal I0 of S0 if j−1(I∗) = I and I0 =I∗∩S0 (equivalently, I∗ =QI0).
Proof. Recall that I∗ = {x ∈ S∗ : there exists H ∈ E such that xH ⊆ I} = {x ∈ S∗ : there exists H ∈ E such that Hx ⊆ I}. Then if I is an ideal so is I∗. In fact, if x∈ I∗ and y ∈ S∗, then there exist F, H ∈ E such that xH ⊆ I and F y ⊆ S. Thus F yxH ⊆ I and it follows that yx∈I∗. Similarly, xy∈I∗ and so I∗ is an ideal. The rest is clear. 2 4. Intermediate extensions
In this section we consider intermediate extensions. Since closed ideals always containt(S) we restrict ourselves to the torsion-free case. So we assume that S is a torsion-free normalizing extension of R.
Recall that if N ⊆ P are submodules of a torsion-free normalizing bimodule, then N is said to be dense in P if [N]P = P. In this case there is a one-to-one correspondence, via contraction, between the set of all the closed submodules of P and the set of all the closed submodules of N ([5], Lemma 2.1). Also, in the torsion-free case we have that
[N]P ={x∈P : there exists H ∈ E such that xH ⊆N}= {x∈P : there exists F ∈ E such that F x⊆N}, ([5], Corollary 4.2).
An intermediate extension is a subring of S containing R. Assume that U ⊆ V are intermediate extensions such that U is dense in V.
If I is an ideal of U, then [I]V is an ideal of V, as is easy to see. Also, if I is a closed R-submodule of U we have I = [I]V ∩U. So if [I]V is an ideal of V, then I is an ideal of U. Thus the following is an obvious extension of Lemma 2.1 from [5].
Lemma 4.1. If U is dense in V, then there is a one-to-one correspondence between the set of all closed ideals of V and the set of all closed ideals of U.
Now we have the following
Proposition 4.2. Assume that U ⊆ V are intermediate extensions and U is dense in V. Then the correspondence of Lemma 4.1 preserves prime and semiprime ideals.
Proof. LetP be a closed submodule of V and put P =P ∩U.
Assume that P is prime and let A,B be ideals of V with AB⊆P. It is easy to see that [A]V[B]V ⊆P, since P is closed. Hence we may suppose that A and B are closed. We have (A∩U)(B∩U) ⊆ P, consequently either (A∩U)⊆ P or (B∩U) ⊆ P and it follows that A = [A]V = [A∩U]V ⊆ [P]V = P or B = [B]V = [B ∩U]V ⊆ [P]V = P. Therefore P is prime.
Conversely, assume thatP is prime andA,B are ideals ofU withAB⊆ P. We have that [A]V[B]V ⊆P and so eitherA⊆[A]V ⊆P orB ⊆[B]V ⊆P. Consequently, A⊆P ∩U =P orB ⊆P ∩U =P and soP is prime.
The proof of the semiprime case is the same with A=B. 2 IfV is an intermediate extension, then [V]S is also an intermediate extension which is closed as an R-submodule of S and V is dense in [V]S. Thus by ([5], Theorem 8.3) there exist a Q-closed submodule V∗ of S∗ such that V∗ ∩S = [V]S and a Z-closed submodule V0 of S0 with V0 = V∗ ∩S0 and V∗ = V0Q. Hence V∗ is a subring of S∗ containing Q and V0
is a saturated subsemigroup of S0 containing Z. By Corollary 8.4 in [5] there is a one-to- one correspondence between the set of all R-closed submodules of V, the set of allQ-closed submodules of V∗ and the set of all Z-closed submodules of V0.
A saturated ideal P0 of V0 is said to be prime (resp. semiprime) if the following holds:
AB⊆P0 (resp. A2 ⊆P), for idealsA, B of V0 (resp. A of P0), implies that eitherA⊆P0 or B ⊆ P0. (resp. A ⊆P0). The semigroup V0 is said to be prime (semiprime) if 0 is a prime (semiprime) ideal of V0.
We have the following extension of Theorem 3.2.
Theorem 4.3. LetV be an intermediate extension ofR, V∗ andV0 as above. Then the one- to-one correspondence between closed submodules gives a one-to-one correspondence between closed ideals (resp. closed prime ideals, closed semiprime ideals).
Proof. By Proposition 4.2 we may assume thatV is closed. Let P be a closed submodule of V, P∗ the extension of P to V∗ and P0 = P∗ ∩V0. As in Theorem 3.2 it follows that when one of the submodules P, P∗, P0 is an ideal, so are the others.
Assume that P is a prime ideal of V and A0, B0 are ideals of V0 with A0B0 ⊆ P0. As in the proof of Proposition 4.2 we may assume that A0 and B0 are closed. Note that (QA0 ∩V)(QB0 ∩V) ⊆ QA0B0 ∩V ⊆ QP0 ∩V = P. Then either A = QA0 ∩V ⊆ P or B =QB0∩V ⊆P and therefore either A0 =A∗∩V0 ⊆P∗∩V0 =P0 or B0 ⊆P0. Thus P0 is prime.
Suppose thatP0 is a prime ideal ofV0 andAB ⊆P∗,A, B ideals ofV∗. As above we may assume that A, B are Q-closed. Then we have (A∩V0)(B ∩V0)⊆ AB∩V0 ⊆P∗∩V0 =P0
and so either A0 =A∩V0 ⊆P0 or B0 ⊆P0. It follows that either A=QA0 ⊆QP0 =P∗ or B ⊆P∗, and thus P∗ is prime.
Finally, assume thatP∗ is a prime ideal of V∗ and AB⊆P, where A, B are ideals of V. Since for x ∈ A∗, y ∈ B∗ there exist F, H ∈ E such that F x ⊆ A and yH ⊆ B, where A∗ (resp. B∗) denotes the extension of [A]V (resp. [B]V) to a closed ideal of V∗ (Corollary 4.8 in [5]), it easily follows thatA∗B∗ ⊆P∗. Hence eitherA∗ ⊆P∗ orB∗ ⊆P∗. Therefore either A⊆A∗∩V ⊆P∗∩V =P or B =B∗∩V ⊆P∗∩V =P and so P is prime.
The semiprime case is the same taking above A0 =B0 (resp. A=B). 2 As a direct consequence of the former results we have the following corollary which holds for any intermediate extension (in particular, for V =S).
Corollary 4.4. Let V be an intermediate extension. Then the following conditions are equivalent:
(i) V is a prime (resp. semiprime) ring.
(ii) V∗ is a prime (resp. semiprime) ring.
(iii) V0 is a prime (resp. semiprime) semigroup.
5. Special types of prime ideals
In this section we study prime ideals of torsion-free normalizing extensions of prime rings.
First, assume thatRis semiprime andM is a torsion-freeR-normalizing bimodule. In ([5], Theorem 4.5) a closed submodule was characterized as a complement submodule. Theorem 2.1 of [4] gives a stronger result for centralizing bimodules. We now give an extension of this result. Note that our proof here is simpler than the proof of [4] for the centralizing case.
Let T be the canonical torsion-free bimodule associated to M ([5], Section 3). Any element x ∈ T can be written as a finite sum x = Pi∈Ωeiai, where (ei)i∈Ω is the set of normalizing generators of L and ai ∈ R are uniquely determined modulo AnnR,r(ei), for all i. The support supp(x) of x is defined as the set of all ei such thateiai 6= 0.
Proposition 5.1. Assume that M is a torsion-free normalizing bimodule over a semiprime ring R and let N ⊆P submodules of M. Then N is closed in P if and only if for any right submodule K of P with N ⊂K there exists 06=x∈K such that RxR∩N = 0.
Proof. One implication is immediate from Theorem 4.5 in [5]. For the other assume thatN is closed inP. First we prove the result forM =T, T as above. Take an elementx∈K\N of minimal support Γ ={e1, . . . , en}, say x=e1a1+e2a2+. . .+enan. If there exists a nonzero element y ∈ N with supp(y) ⊆ Γ we may assume that y(e1) 6= 0, where y(e1) denotes the e1-coefficient of y. Let I be the ideal of R of all the elementsa such that there exists z ∈N with supp(z)⊆Γ and z(e1) =a. Then I is a nonzero ideal of R and H =I⊕Ann(I) is an essential ideal.
For any 06=b ∈I there exists z =e1a1b+e2b2+. . .+enbn∈N and we have xb−z ∈K and supp(xb−z) ⊂ Γ. By the minimality of supp(x) we have xb−z ∈ N and so xb ∈ N. Since x /∈N and N is closed, xH 6⊆N. Therefore there existsc∈Ann(I) such that xc6= 0.
Hence xc∈K and RxcR∩N = 0.
Now the general case can be proved in a canonical way using the epimorphism Φ :T →M.2 Note that if R is primeE is the set of all nonzero ideals of R.
Lemma 5.2. If R is prime and M is a normalizing torsion-free bimodule over R, then there exists a submodule L of M which has a normalizing free basis and is dense in M.
Proof. If x ∈ M is a normalizing element and xa = 0, a ∈ R, then xRaR = RxaR = 0 and sox= 0, since M is torsion-free. Let (xi)i∈Ω be the set of normalizing generators of M. Then there exists a maximal right R-independent set of generators E = (xi)i∈Λ. If L is the free submodule of M generated byE, then it is easy to see that Lis dense in M. 2 The free submodule L of M will be called afree dense submodule of M.
Corollary 5.3. If R is prime and M is a normalizing bimodule over R, then the canonical torsion-free extension M∗ of M is free over Q.
Proof. LetL be a free dense submodule of M/T(M). It is easy to see that L∗ is free over Q
and that M∗ = (M/T(M))∗ =L∗. The result follows. 2
In the rest of the paperS is always a torsion-free normalizing extension of a prime ringR and V is an intermediate extension. If I is an R-disjoint ideal of V, then [I]V is also R-disjoint.
Moreover, ifI∩R 6= 0, then [I]V =V. Hereafter we denote by [I] the closure [I]V of I inV. Now we extend and improve results of ([2], Section 6). Recall that a ring T is said to be (right) strongly prime if any nonzero ideal J of T contains a (right) insulator, i.e., a finite setF ⊆J such that AnnT,r(F) = 0.
Also the (right) singular ideal Z(T) of a T is the set of all the elements x∈T such that AnnT,r(x) is an essential right ideal of T ([6], pag. 30–36). The ring T is said to be (right) non-singular if Z(T) = 0.
In the following strongly prime (non-singular) means right strongly prime (right non- singular). An idealP of T is said to be strongly prime (non-singular prime) if the factor ring T /P is strongly prime (non-singular prime).
Proposition 5.4. Let R be prime ring and V be intermediate extension, as above. Then P is a closed prime ideal of V provided that one of the following conditions is fulfilled:
(i) P is an ideal of V which is maximal with respect to P ∩R = 0.
(ii) P is a strongly prime R-disjoint ideal of V.
Proof. (i) SinceR is prime it follows easily that P is prime. Also, since P ∩R = 0 we have [P]∩R = 0 and maximality of P implies that P = [P]. Hence P is closed.
(ii) Suppose that [P]⊃P. Then there exists a finite set F ⊆[P] such thatF x⊆P, x∈V, implies x ∈ P. However, since F is finite, there exists H ∈ E with F H ⊆ P and H 6⊆ P.
This is a contradiction and the result follows. 2
The following result is an extension of Theorem 6.2 in [2].
Theorem 5.5. Assume thatR is a non-singular prime ring andV an intermediate extension.
If P is an ideal of V which is maximal with respect to P ∩R = 0, then P is a non-singular prime ideal.
Proof. Assume, by contradiction, that Z(V /P) =I/P 6= 0, where I is an ideal of V. By the maximality ofP there exists 06=a∈I∩R. We show thata∈Z(R), which is a contradiction.
Take a free dense submodule L of S with basis (ei)i∈Λ over R. For a nonzero right ideal J of R the right ideal N =J V +P of V properly contains P. Thus K ={y ∈N :ay ∈P} is a right ideal of V with P ⊂ K because a+P ∈ Z(V /P). Since P is closed we easily get P ∩L⊂K∩L.
Note that ifx∈J V we can write x=Piaivi, whereai ∈J and vi ∈V. Take a nonzero ideal H of R such that viH ⊆ L, for any i. We can easily see that for any h ∈ H we have xh∈P⊕jJ ej. It follows that xh can uniquely be represented in the basis (ei)i∈Λ with coefficient inJ.
By the above there exists an elementz ∈J V ∩L\P such thataz ∈P. SinceP is closed, changing z by zh we may assume that z /∈ P and it can be represented as an element of
P⊕jJ ej, say z =Pnj=1bjej with 06=bj ∈J, for 1≤j ≤n, and {e1, . . . , en} is minimal.
We claim that do not exist 0 6= y ∈ P ∩L such that supp(y) ⊆ {e1, . . . , en}. In fact, let 0 6= y = a1e1 +. . .+anen ∈ P ∩L. We may suppose that a1 6= 0. For any r ∈ R there exists r0 ∈ R with e1r = r0e1. Also there exists 0 6= a0 ∈ R with e1a0 = a1e1. Thus zr0a0−b1ry∈P⊕jJ ej,a(zr0a0−b1ry)∈P and supp(zr0a0−b1ry)⊂ {e1, . . . , en}. Therefore zRa0R⊆P, which is a contradiction since P is closed.
By the claim we have that az = 0, since az ∈ P and supp(az) ⊆ {e1, . . . , en}. Conse-
quently AnnJ,r(a)6= 0 and the proof is complete. 2
Now we prove the converse of Theorem 5.5.
Proposition 5.6. Let R be a prime ring and V an intermediate extension. If P is an R-disjoint non-singular closed prime ideal ofV, then R is non-singular.
Proof. Suppose that 06=a∈Z(R) and let K be a right ideal of V with K ⊃P. Take a free dense submodule L of S with basis (ei)i∈Λ overR. Since P is closed, K∩L⊃P ∩L. Then there exists x = Pnj=1ajej ∈ K∩L\P ∩L of minimal support with this property, i.e., for any element y ∈K∩L such that supp(y)⊂ supp(x) we have y∈P ∩L. As in the proof of Theorem 5.5 we show that for any y∈P ∩Lwith supp(y)⊆supp(x) we have that y= 0.
Since a1R 6= 0 there exists r ∈ R such that a1r 6= 0 and aa1r = 0. Also, let r0 ∈ R be with e1r0 = re1. Thus 0 6= xr0 ∈ K ∩L and so axr0 = 0, since supp(axr0) ⊂ supp(x).
Hence AnnK/P,r(a)6= 0 and we have a+P ∈ Z(V /P) = 0. Consequently a∈ P ∩R = 0, a
contradiction. This shows that Z(R) = 0. 2
Recall that if S is a strongly normalizing extension of R and P is a prime ideal of S, then P =P ∩R is a prime ideal of R ([10], Proposition 1.5). Also, if I is an ideal of R, then IS is an ideal of S with IS∩R = I. Thus, by factoring out the ideals P and PS from R and S, respectively, we immediately have the following
Corollary 5.7. Let R be a prime ring, S a strongly normalizing extension of R and V an intermediate extension. If P is a prime ideal of R and P is an ideal of V which is maximal with respect to P ∩R = P, then P is non-singular in R if and only if P is non-singular in V.
Now we consider strongly prime rings and ideals. The following result is an extension of Theorem 6.1 of [2].
Theorem 5.8. Let R be a strongly prime ring, V an intermediate extension and P an ideal of V which is maximal with respect to P ∩R = 0. Then P is a strongly prime ideal.
Proof. Suppose that I is an ideal of V with I ⊃ P. Then I∩R 6= 0 and so there exists a finite set F ⊆I∩R such that AnnR,r(F) = 0. PutK ={y∈V :F y ⊆P}, a right ideal of V containingP. We prove that K =P and this shows that F is an insulator in V /P.
Assume, by contradiction, that K ⊃ P. By Propositions 5.4 and 5.1, there exists 0 6=
x ∈ K such that RxR∩P = 0. Let L be a free dense submodule of S. Then there exists a nonzero ideal H of R such that xH ⊆ L and we have F xH ⊆ RxR∩P = 0. We easily obtain xH = 0, since Lis free. This is a contradiction because S is torsion-free. 2 The converse of Theorem 5.8 holds if we assume that S is a strongly normalizing extension of R.
Proposition 5.9. Let S be a torsion-free strongly normalizing extension of R and V an intermediate extension. If P is a strongly prime R-disjoint ideal of V, then R is strongly prime and P is closed.
Proof. Let H be a nonzero ideal of R. Then V HV is a nonzero ideal of V and V HV 6⊆P. Thus there exists a finite setF ⊆V HV such that F x⊆P,x∈V, implies x∈P. Moreover, any yj ∈F ⊆V HV ⊆ S can be written as yj = Pixiaij, for aij ∈ H, since S is a strongly normalizing extension, where (xi)i are strongly normalizing generators. Therefore,{aij} ⊆H is an insulator in R and so R is strongly prime. Finally, P is closed by Proposition 5.4. 2 As in Corollary 5.7 we immediately have the following.
Corollary 5.10. Let S be a strongly normalizing extension of R and V an intermediate extension. If P is an ideal of R and P is an ideal of V which is maximal with respect to P ∩R=P, then P is strongly prime if and only if P is strongly prime.
Now we relate strongly primeness between S, S∗ and S0. We say that a subsemigroupV0 of S0 is strongly prime if any nonzero ideal of V0 contains an insulator, i.e. a finite subset F0 with AnnV0,r(F0) = 0.
Note that any ideal of V0 is closed and so is saturated. In fact, a nonzero ideal of Z has a nonzeroR-normalizing element of Q, so contains an invertible element ofQ. Thus, ifI is a nonzero ideal of V0 and for z ∈V0 we have zH ⊆I, where 06=HZ, it follows that z ∈I.
First we prove the following
Lemma 5.11. Let S be a torsion-free strongly normalizing extension of R and assume that U ⊆ V are intermediate extensions such that U is dense in V. Then U is strongly prime if and only if V is strongly prime.
Proof. If U is strongly prime and I is a nonzero ideal of V, then J =I∩U 6= 0. Thus there exists a finite set F ⊆J such that AnnU,r(F) = 0. We can easily see thatAnnV,r = 0 and so V is strongly prime.
Conversely, assume thatV is strongly prime and letI be a nonzero ideal ofU. Then [I]V is a nonzero ideal ofV. Thus there exists a finite setF ⊆[I]V such thatAnnV,r(F) = 0. Also there exists a nonzero ideal H of R such that HF ⊆ I. By Proposition 5.9, R is strongly prime. Hence there exists a finite setF0 ⊆Hwith AnnR,r(F0) = 0. ConsequentlyF0F ⊆I is a finite set which is an insulator in U. In fact, take x∈U such thatF0F x= 0. By Corollary 5.3 the canonical torsion-free normalizing extension S∗ of S is free over Q. Using this we
easily see that F x= 0. Thereforex= 0 and we are done. 2
Remark 5.12. A similar argument as in Lemma 5.11 shows that if S is a normalizing extension, R is strongly prime and V is an intermediate extension, for an ideal I of V there exists an insulator in I if and only if there is an insulator in [I]V. Thus to show that V is strongly prime it is enough to find an insulator in any nonzero closed ideal of V.
To end the paper we prove the following
Theorem 5.13. Let R be a prime ring, S a torsion-free strongly normalizing extension of R and V an intermediate extension. Let V∗ and V0 be the corresponding closed subrings of S∗ and S0 with V∗∩S = [V]S and V0 =V∗∩S0. The following conditions are equivalent:
(i) V is strongly prime.
(ii) R and V∗ are strongly prime.
(iii) R and V0 are strongly prime.
Proof. We may assume that V is closed in S, by Lemma 5.11.
(i)→ (iii) Suppose that V is strongly prime. ThenR is strongly prime by Proposition 5.9.
Let I0 be a nonzero ideal of V0. Then we have that I = QI0 ∩V is a nonzero ideal of V and so there exists a finite set F ={y1, . . . , yn} ⊆ I such that AnnV,r(F) = 0. For any i we can write yi = Pjqijmij, qij ∈ Q, mij ∈ I0. We can easily see that F0 = {mij} ⊆ I0 is an insulator in V0, since S∗ is free as right R-module.
(iii) → (ii) Let I be a nonzero ideal of V∗. We show that I contains an insulator. Since R is strongly prime we have that Q is also strongly prime. So by Remark 5.12 we may assume that I is closed in V∗. Put I0 = I ∩V0, a nonzero ideal of V0. So there exists a finite set F ⊆I0 such that AnnV0,r(F) = 0. It is not hard to show that F ⊆I is an insulator in V∗. (ii) → (i) Let I be a nonzero closed ideal of V. Then there exists a closed ideal I∗ of V∗ with I =I∗∩V. By assumption there exists a finite set F ⊆ I∗ such that AnnV∗,r(F) = 0.
Also, since F is finite F H ⊆ I, for some 0 6= H R. Take a finite set F0 ⊆ H such that
AnnR,r(F0) = 0. Then F F0 ⊆I is an insulator in V. 2
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Received July 8, 2002; revised version January 29, 2003
